Problem: You have found the following ages (in years) of 4 seals. Those seals were randomly selected from the 38 seals at your local zoo: $ 8,\enspace 7,\enspace 3,\enspace 11$ Based on your sample, what is the average age of the seals? What is the standard deviation? You may round your answers to the nearest tenth.
Answer: Because we only have data for a small sample of the 38 seals, we are only able to estimate the population mean and standard deviation by finding the sample mean $({\overline{x}})$ and sample standard deviation $({s})$ To find the sample mean , add up the values of all $4$ samples and divide by $4$ $ {\overline{x}} = \dfrac{\sum\limits_{i=1}^{{n}} x_i}{{n}} = \dfrac{\sum\limits_{i=1}^{{4}} x_i}{{4}} $ $ {\overline{x}} = \dfrac{8 + 7 + 3 + 11}{{4}} = {7.3\text{ years old}} $ Find the squared deviations from the mean for each sample. Since we don't know the population mean, estimate the mean by using the sample mean we just calculated {0.49} + {0.09} + {18.49} + {13.69}} {{4 - 1}} $ {s^2} = \dfrac{{32.76}}{{3}} = {10.92\text{ years}^2} $ As you might guess from the notation, the sample standard deviation $({s})$ is found by taking the square root of the sample variance $({s^2})$ ${s} = \sqrt{{s^2}}$ $ {s} = \sqrt{{10.92\text{ years}^2}} = {3.3\text{ years}} $ We can estimate that the average seal at the zoo is 7.3 years old. There is also a standard deviation of 3.3 years.